This is the right answer. A large institution like a university is not a monolith. The university will produce antibodies to external participants, like lawyers (or the media). You're most likely to get better outcomes if you can ensure the conflict plays out within the university and its rules, structures, and participants. Your work now is to convince other members of the university to advocate on your behalf. (It should not be difficult. If this is as you describe, reasonable faculty members will be your allies.)

The same goes for publicizing this further. The student newspaper is probably okay, but keep talking to other media in the room as a bargaining chip. Bad press may well force some administrator's hand.

To be sure, a chat with a lawyer may be helpful to get a sense of the universe of possible outcomes pursuing extramural action, but don't let anbyody send any lawyerly letters yet.